source: https://doi.org/10.7892/boris.142910 | downloaded: 1.2.2022
Coordination Failure and Financial Contagion
Michael Manz
02-03 March 2002
Diskussionsschriften
Universität Bern
Volkswirtschaftliches Institut Gesellschaftstrasse 49 3012 Bern, Switzerland
Coordination Failure and Financial Contagion
Michael Manz
∗University of Bern
March 2002
Abstract
This paper explores a unique equilibrium model of ”informational”
Þnancial contagion. Extending the global game model of Morris and Shin (1999), I show that the failure of a singleÞrm can trigger a chain of failures merely by affecting the behavior of investors. In contrast to the existing multiple equilibria models of Þnancial and banking pan- ics, there is no indeterminacy in the present model. Thus, it provides a clear framework to assess the consequences of contagion and yields some important and hitherto unnoticed insights. Most importantly, if contagion is compared to an appropriate benchmark, its impact can be both positive or negative, which contrasts sharply with the traditional view of contagion. Moreover, contagion increases the correlation be- tween Þrms, but the effect on the unconditional probability of failure is exactly zero.
Keywords: Þnancial contagion, systemic risk, Þnancial crises, global games, unique equilibrium.
JEL-ClassiÞcations: G15, G21, C72
∗University of Bern, Institute of Economics, Vereinsweg 23, CH-3012 Bern, Switzer- land, michael.manz@vwi.unibe.ch. I thank Ernst Baltensperger, Simon Loertscher, Esther Bruegger, Manuel Waelti and Alain Egli for helpful discussions.
1 Introduction
As is well known in the growing literature on Þnancial contagion and sys- temic risk, there are at least two potential channels through which a single Þrm can affect other Þrms. The Þrst channel rests upon on direct capital connections between the Þrms, which make it possible that the failure of a debtor leads to the failure of its creditors simply because the latter have to write offtheir claims. While it is difficult to get an accurate picture of these credit linkages in practice, the mechanism of this transmission channel seems to be well understood. Recent theoretical models which deal with such con- tagion in a banking framework include Rochet and Tirole (1996), Allen and Gale (2000) and Dasgupta (2001). A second channel, which I will refer to as
”informational contagion”, hinges on the beliefs of Þnancial market partici- pants. In a banking context, depositors (or also other banks) may decide to withdraw their claims when observing the collapse of another bank because they lack precise information on how the failure is related to their own bank.
Likewise, international investors might withdraw their investments from a country when they observe trouble in a different country, and due to such reactions of investors, a crisis can spread from one Þrm or country to others.
Since the present paper focuses on this second channel, I will refer to infor- mational contagion whenever I speak of (Þnancial) contagion hereafter.1,2
Although informational contagion is widely believed in3 and is a major concern behind regulatory measures such as introducing a deposit insurance, providing implicit government guarantees or imposing capital controls, there are few formal models which deal with the informational propagation channel.
Recent exceptions include Chen (1999) and Aghion, Bolton and Dewatripont (2000). Obviously, a major impediment to a sound theoretical foundation of informational contagion is the presence of multiple equilibria in the usual
1Typically, contagion is either discussed in a banking or in an international crises context, so I will also have these examples in mind, although my model is rather general.
Since the literature is far from converging to a common deÞnition of the widely used terms contagion and systemic risk, I renounce to give a precise deÞnition until section 4.
2For a discussion of different bank contagion channels, see Kaufman (1994). Dornbusch et al. (2000) survey international contagion channels.
3See e.g. Park (1991), who argues that the lack of bankspeciÞc information is the main source of bank contagion and offers evidence in favour of this view. Calomiris and Mason (1997), and Gorton (1988) alsoÞnd evidence of informational bank contagion. For evidence on informational contagion across countries, see Ahluwalia (2000), Park and Song (2000), and also Eichengreen, Rose and Wyplosz (1996).
coordination failure and bank run models as in Diamond and Dybvig (1983).
Since the events which determine the beliefs of the depositors and may trigger a run are not part of these models, it remains open which equilibrium will occur. Hence, there is an apparent indeterminacy in any sunspot model. If theory offers little guidance on the circumstances which trigger a single bank run, it seems even less suited to predict in which situations contagion will occur. Put differently, if a model ”predicts” that depending on the agents’
beliefs, any outcome ofÞrmAcan be an equilibrium, but remains completely silent about the beliefs, it is hardly able to predict if and how the outcome of ÞrmA could affect a different ÞrmB. The same problems arise in models of international Þnancial crises with self-fulÞlling features in the manner of Obstfeld (1996).
There is, however, a recent strand of literature, initiated by Carlsson and van Damme (1993) and further advanced by Morris and Shin (1998) in a speculative attack framework, which has developed a technique that selects a unique equilibrium in many coordination failure models.4 The common feature of this ”global games” approach is that the payoffs of the players depend on some random state of the world, which is however not common knowledge. Instead, each player privately observes a noisy signal of the true state. Interestingly, this assumption is not only a step towards a more realistic information structure, but also introduces a useful structure into the beliefs of agents which leads a unique equilibrium. In another contribution, Morris and Shin (1999) provide a model of a coordination failure among numerous small creditorsÞnancing a common project. They show that in the presence of noisy but sufficiently precise private information of the creditors, there is a unique threshold of the realization of some fundamental variable below which the borrower is ”run” or denied sufficient credit such that the project fails. Although systemic risk is neither an issue in Morris and Shin (1999) nor in any other model with only one Þrm, there is little doubt that Carlsson and van Damme’s (1993) unique equilibrium selection technique provides a promising framework to analyze contagion.
In this paper, I therefore extend a modiÞed version of the Morris and Shin (1999) model to include two (or more)Þrms and two (kinds of) fundamental
4A precursor of this literature is Rubinstein (1989). Morris and Shin (2001) provide a good survey of this ”global games” approach. Goldstein and Pauzner (2000) apply the technique to the Diamond and Dybvig (1983) model. Rochet and Vives (2000) also explore a banking model. Finally, it shall be mentioned that an earlier bank model with a unique Bayesian equilibrium is found in Postlewaite and Vives (1987).
variables. TheÞrst fundamental represents states which are speciÞc for each Þrm, while the second fundamental is a common state inßuencing all Þrms within an industry or region. Further, I introduce a time structure such that the investors of the second Þrm can observe what happens to the Þrst project before deciding whether to roll over their loans. In this model, there is still a unique equilibrium in which eachÞrm succeeds if and only if theÞrm speciÞc fundamental exceeds some threshold, which depends on the state of the common fundamental. In addition, it can be shown that the equilibrium threshold of the second Þrm is larger when theÞrst Þrm fails, which implies that a failure of the Þrst Þrm increases the probability that the second Þrm fails. Hence, there is a contagious link between both projects which stems from a purely informational channel. If the creditors of the second Þrm observe that the Þrst Þrm fails, they adjust their prior beliefs on the state of the common fundamental accordingly. As a consequence, they become more reluctant to roll over their loans, which in turn may cause the collapse of the secondÞrm. On the other hand, by the same mechanism, a good result of the ÞrstÞrm can rescue the other, which is a positive aspect of contagion which - as far as I know - has received no attention in the literature so far. Finally, the outcome of the model can be compared to a natural benchmark in which there is no contagion. As an important result, if we are only concerned about the unconditional likelihood of failure of theÞrms, the net effect of contagion is exactly zero. This result emerges despite the fact that a failure of the ÞrstÞrm can trigger the failure of the second Þrm (and possibly moreÞrms).
Therefore, the model provides not only a rigorous theoretical foundation of an informational channel of contagion, but also reveals that the implications of such contagion are quite different from those which the traditional view of contagion suggests.
The present work is probably most closely related to the recent contri- butions of Dasgupta (2001) and Goldstein and Pauzner (2001), which both analyze the issue of contagion in a global game framework. However, the propagation mechanisms in their models are different. Dasgupta (2001) de- velops a banking model in which contagion stems from the existence of capital links between Þnancial institutions. Goldstein and Pauzner (2001) explore a model with two countries subject to self-fulÞlling Þnancial crises, in which contagion emerges due to wealth effects. In their framework, a crisis in one country reduces investors’ wealth, which makes them more risk averse and more inclined to withdraw their investments from the second country. This in turn raises the likelihood of a crisis in the second country. The model pre-
sented in this paper, by contrast, presumes neither capital links nor wealth effects. To my knowledge, it is the Þrst model of informational contagion in a global game framework. Since the consideredÞrms need not be banks, the model is rather general, though banks seem to be typical examples of bor- rowers facing numerous small creditors. If the differentÞrms are interpreted as representative Þrms of different countries, the model can also be applied in an international setting to explore why and how different countries are affected when one country experiences a Þnancial crisis.
The remainder of the paper is structured as follows. In section 2, I present the basic structure of the model and derive the unique equilibrium of the game. For presentation purposes, the model is introduced with only one Þrm, while the secondÞrm is introduced in the extended model presented in section 3. This part of the paper also contains the important result that there is a contagious link between theÞrms. In section 4, I discuss the meaning and the consequences of Þnancial contagion in some more detail and deÞne the concepts of ”negative” and ”positive” contagion, while section 5 concludes.
2 The Basic Model
2.1 Structure of the Model with one Firm
To begin with and for ease of exposition, suppose there is only one Þrm, which runs a project that is Þnanced by a continuum of creditors normalized to one.5 In section 3, I will extend the model to include two or more Þrms, which is of course a crucial step for the analysis of contagion. The basic structure of the model and many features of the equilibrium, however, can well be discussed in the framework with one Þrm. The model focuses on the behavior of the individual creditors, who, ad an ad interim stage of the project, receive the opportunity to rethink their investment and to decide on whether to roll over their loans. The initial lending decision is not part of the model, yet assume that the loan conditions are such that it is ex ante rational to lend. Together with the realizations of two economic fundamental variables, the fraction l of creditors who decide to withdraw their loans in the intermediate period determines whether the Þrm succeeds or fails. At a Þnal stage, the project is terminated and yields a gross return of R ≥1if it
5The project represents the whole and only activity of the Þrm, hence I will use the termsÞrm and project interchangeably.
succeeds, whereas the return is 0 if it fails.6
The returnr(θ, u, l)of the project is speciÞed as r(θ, u, l) =
( R if θ> u+zl
0 if θ ≤u+zl , (1)
where θ and u index the states of two independent economic fundamentals.
More speciÞcally, assume that the state θ is drawn according to a uniform distribution on the unit interval, while the state u is either u1 or u2 with probabilities P(u = u1) = wa and P(u = u2) = 1−wa, where u1 > u2. Notice from (1) that strong economic fundamentals are associated with a high θ and a low u. Of course, as long as only one project is considered, it seems unnecessarily complicated to introduce two fundamentals. However, the role of the second fundamentaluwill become clear in the model with two Þrms in section 3, where I will assume that the state u equally affects both Þrms, whileθ is related to a speciÞcÞrm. That is,θ summarizes factors that affect only one Þrm and could for example represent the ex ante unknown quality of the hired chief manager, whereas u could be a proxy variable for the demand facing the products of both Þrms.
The parameter z > 0 in equation (1) captures the disruption caused by early loan withdrawals, which can be due to the Þrm having to sell illiquid long-term assets at aÞre-sale loss in order to pay out the lenders. The higher z, the more harm is caused by withdrawing funds before the project matures.
Further, assume that
u2 >0, u1+z <1, (2) which implies that theÞrm speciÞc fundamentalθis in some sense dominant, because a sufficiently high (low) state θ can always save (ruin) the project.
Technically, assumption (2) guarantees the existence of a lower and an upper
”dominance region” of θ for which agents have dominant actions. There is now a tripartite classiÞcation of the fundamental variables. If the state of fundamentals turns out to be very poor, i.e. if θ ≤ u, the project fails even under the best Þnancing conditions where nobody withdraws funds at the intermediate state. Conversely, ifθ > u+z,the project succeeds irrespective
6The date of termination may well depend on whether the project succeeds. In many cases, including all kinds of runs, it seems plausible to assume that the project must be abandoned immediately after the intermediate period if it is a failure, while otherwise, it can be orderly terminated at its time of maturity.
of the actions of the creditors, whereas if u <θ ≤u+z, the outcome of the project depends in a crucial way on the actions of the creditors.
Creditors have Von Neumann - Morgenstern preferences and are risk neu- tral. Hence, they maximize expected payoffs. If rolling over the loan, they receive the face value of debt normalized to 1 if the projects succeeds and nothing otherwise. If they withdraw early, they obtain some amount λ with certainty, where0<λ<1.One interpretation of this payoffstructure is that lenders can liquidate a collateral which yields λ in the intermediate period but becomes worthless if the project fails. Alternatively, the difference 1−λ may represent an interest rate which is paid to patient lenders, provided the project succeeds. The payoffs of the creditors can be summarized as follows:
θ≤u+lz θ > u+lz
Roll over 0 1
Withdraw λ λ
It is important to see that if the states ofθanduwere common knowledge, the model would result in multiple equilibria whenever u ≤ θ < u+z . In this case, if everyone rolls over the loan, no single agent has an incentive to deviate given the others’ strategies, since the project succeeds anyway.
On the other hand, if nobody rolls over, nobody has an incentive to deviate either, because the project certainly fails. Therefore, even if one neglects mixed strategies, there are at least two obvious Nash equilibria. However, these conclusions change considerably if we put the game in a ”global game”
framework with incomplete information.
Assume, therefore, that the distributions of the fundamental variables are common knowledge, but the true state of the fundamentals is not observable.
Instead, before deciding whether to roll over the loan, each creditoriprivately observes a noisy signal
si =θ+εi (3)
on the state θ, where εi is uniformly distributed on [−ε,ε] and independent of θ and εk∀k 6=i. Assume further that
2ε≤min[u2,1−u1−z], (4) which means that the signal is at least of some minimal precision. One natural interpretation of this framework with noisy signals is that investors have access to different sources of information. Alternatively, they might slightly differ by their interpretation of publicly available information.
Finally, an important assumption of the model is that the creditors can- not coordinate their behavior, i.e. they cannot meet to share their private information - in which case they could learnθ by a law of large numbers - or to coordinate their action before deciding on their credit. This assumption is certainly plausible for depositors of a bank or other small and dispersed investors. Thus, the decision whether to withdraw funds or not is the result of a noncooperative game among the creditors. Moreover, since the type of each player, which is characterized by the signal si, is private knowledge, it is a game of incomplete information. In the next section, I show that this game has a unique Bayesian Nash equilibrium.
2.2 Unique Equilibrium
Creditors maximize expected payoffs. Hence, they roll over their loan if and only if the expected payoff from doing so exceeds the opportunity costs λ.7 The only available information upon which this decision can be conditioned is the signalsi, so a strategy of a creditor is a plan which maps each signalsiinto one of the two actions ”withdraw” respectively ”roll over”. In an equilibrium proÞle of strategies, each creditor’s strategy maximizes her expected payment given that the other creditors follow the strategies in the proÞle.
Before turning to the main results of this section, consider some prelim- inaries. Since the error term εi of the signal is drawn from a uniform distri- bution, the posterior belief θ | si of creditor i is also uniformly distributed, that is:
θ|si
∼u [si−ε, si+ε]. (5) More precisely, this is only true if ε≤si ≤1−ε,but given the assumption in (4), the optimal response to any signal belowεor above1−εis trivial anyway.
To see this, note that any creditor who observes a signal si < s = u2 −ε knows that θ< u2, which implies that the project fails even in the best case where u = u2 and l = 0. Hence, the payoff from rolling over the loan is zero with certainty, and the lender will withdraw her loan. Conversely, any signal si > s=u1+z+ε implies θ> u1+z, such that the project succeeds even in the worst case where u =u1 and l = 1, and so the loan is certainly not withdrawn. To summarize, any rational creditor withdraws if observing a signal si < s and rolls over whenever si > s. Hence, from here on, I will
7Where it is implicitly assumed that they terminate the loan if the expected payoff from both actions is exactly equal. This assumption is not crucial for any results.
restrict attention to signals in the interval [s, s].Since the assumption in (4) guarantees that s ≥ε and s≤ 1−ε, the distribution given in (5) is correct for all relevant si ∈[s, s].
What is now the optimal strategy of a creditor who receives a signal between s and s? Notice that the role of the signal is twofold. First, the signal contains (possibly very precise) information on the state θ. Second and perhaps more importantly, the signal is known to be correlated with the private signals of others, which allows an inference regarding their beliefs and actions. Unlike in a setup where θ is common knowledge, it is no longer the case that any beliefs about the beliefs of others are equally permissible.
The signal structure, therefore, serves as a device to coordinate the (higher order) beliefs of agents. It is exactly this role of the signal, together with the existence of dominance regions ensured by assumption (2) and the strategic complementarity proved in appendix 1, which leads to a unique equilibrium of the game. Intuitively, observing a high signal is good news for each creditor, because it means that the state θ is good. Since observing a high signal also implies that the others have observed a high signal (and that all believe that all have observed a high signal, and so on), it makes each creditor believe that the other lenders are more inclined to roll over their loans. This is again good news and increases the incentives to roll over the loan even further. Hence, one might conjecture that there is some thresholdsof the signal above which the loan is rolled over. In the remainder of this section, I will show that this guess turns out to be exactly true and that the following result holds:
Proposition 1 The game among creditors has a unique equilibrium in which there is a threshold θ∗(u) of θ, depending on the state u of the other funda- mental, such that the Þrm succeeds if and only if θ >θ∗(u).
The proposition can be proved in two main steps which are stated in lemma 1 and 2. To begin with, suppose that all lenders follow a switching or monotone strategy around some threshold parameters. Thus, depending on the observed signal si, each creditor i takes either action0 (withdraw) or 1 (roll over). The switching strategy arounds can be formalized as:
Is(si) =
( 0if si ≤s 1if si > s .
Further, denote byQ(s, Is)the expected payoff from rolling over the loan of a creditor who observes the signal si =s and believes that all other creditors follow strategy Is. We can then derive the following result:
Lemma 1 If all creditors follow a switching strategy around some threshold s, there is a unique value ofs which solves Q(s, Is) =λ.
In order to prove lemma 1, it is sufficient to establish thatQ(s, Is) raises from 0 to 1 and is strictly increasing in s over the whole range [s, s]. This is what I now show. If every lender applies the switching strategy Is, the proportion of creditors terminating the loan in the intermediate period con- ditional on s andθ is
l(s,θ) =P(si ≤s| θ) =
Z min[s,θ+ε,]
min[θ−ε,s] (2ε)−1dsi
respectively l(s,θ) =
0 if s <θ−ε
1
2ε[s−(θ−ε)] if θ−ε≤s≤θ+ε
1 if s >θ+ε
(6) For both realizationsu∈{u1, u2},there is a unique critical value ofθ, solving
θ =u+zl(s,θ),
above which the project succeeds. Given equation (6) and conditional on u∈{u1, u2}, this threshold of θ solves
θ=
u if s <θ−ε
u+2εz [s−(θ−ε)] if θ−ε≤s≤θ+ε u+z if s >θ+ε.
(7) Let θcrit(s, u) denote this critical threshold of θ, conditional on s and on u∈{u1, u2}. Hence, solving (7) for θ, θcrit(s, u) is determined as
θcrit(s, u) =
u ifs < u−ε
2εu+z(s+ε)
2ε+z if u−ε≤s ≤u+z+ε u+z if s > u+z+ε
(8) Of course, if the state u were known, no s below u−ε or above u+z +ε could ever be an optimal switching threshold, but the creditors mustÞnd their optimal strategy without knowingu.Conditional onu∈{u1, u2}, and given
the posterior distribution of θ deÞned in (5), the expected payoff Q(s, Ie s, u) from rolling over the loan is
Q(s, Ie s, u) =P(θ ≥θcrit(s, u)|si =s) =
Z max[s+ε,θcrit(s,u)]
max[s−ε,θcrit(s,u)]
(2ε)−1dθ.
From equation (8), Q(s, Is, u) can then be computed as Q(s, Ie s, u) =
0 if s < u−ε
1
2ε+z[s+ε−u] if u−ε≤s≤u+z+ε 1 if s > u+z+ε
(9)
GivenQ(s, Ie s, u)for both statesu∈{u1, u2}, it is straightforward to compute Q(s, Is). To keep the notation general enough for later purposes, I denote in the following by w the probability that u is u1, conditional on all available information. Of course, in the model with one Þrm, w is simply the uncon- ditional probability wa, but in section 3, w will depend on some additional information of the creditors. Since the best guess ofP(u=u1)is given byw, the unconditional expected payoff Q(s, Is) of the marginal creditor is then
Q(s, Is) =wQ(s, Ie s, u1) + (1−w)Q(s, Ie s, u2).
Together with equation (9), this gives us an expression for Q(s, Is)in terms of the model parameters. Notice that Q(s, Is) = 0 for all s < s = u2 −ε and that Q(s, Is) = 1 for all s > s = u1 +z +ε. Furthermore, as long as s < u1−ε,Q(s, Ie s, u1)is zero, and for all s > u2+z+ε, Q(s, Ie s, u2) is one.
Therefore, assuming u1−ε ≤ u2 +z +ε,8 the range [s, s] can be split into three parts, and the expected payoff from rolling over the loan is:
Q(s, Is) =
(1−w)[s+ε−u2]
2ε+z if s≤s < u1−ε
s+ε−[wu1+(1−w)u2]
2ε+z if u1−ε≤s≤u2+z+ε
w[s+ε−u1]
2ε+z + (1−w) if u2+z+ε≤s ≤s.
(10) Figure (1) plots the graph of Q(s, Is), which is characterized by three linear intercepts with different slopes, for an arbitrary parameter setting. Although
8This assumption ensures that the intermediate range is at least of measure zero. Oth- erwise, ifu1−ε> u2+z+ε,there would be a range in whichQ(s, Is)is1−wand we would have to assumeλdoes not take the particular value1−wto guarantee the uniqueness of the equilibrium.
the slope changes twice,Q(s, Is)is continuous insover the whole range[s, s]
and the derivative with respect to s is
∂Q(s, Is)
∂s =
1−w
2ε+z if s < s < u1−ε
1
2ε+z if u1−ε< s < u2 +z+ε
w
2ε+z if u2+z+ε< s < s
Clearly, this derivative is strictly positive.9 Thus Q(s, Is) runs from 0 to 1 and is monotonically increasing in s ∀ s∈[s, s], which implies that for each λ, there is exactly one value of s which solves Q(s, Is) =λ, as is illustrated in Þgure (1). This proves lemma 1.
Figure 1: Threshold s∗ for (u1, u2, z,ε,λ, wa)=(0.4,0.2,0.5,0.05,0.3,0.5) So far, I have restricted attention to pure switching strategies in which creditors roll over their loan if and only if the signal exceeds some threshold, though in principal they might also follow mixed strategies, presumably giv- ing more weight to the action ”roll over” if the signalsiincreases.10 However,
9The fact that the derivative is not deÞned ifsis eithers, u1−ε,u2+z+εorsdoes not change the conclusion that Q(s, Is)is strictly increasing insin the range[s, s].
10Some authors restrict attention to such switching strategies from the beginning. See for instance Dasgupta (2001) and Rochet and Vives (2000).
the following lemma states that the unique equilibrium strategy of the game is indeed a pure switching strategy around the threshold s∗.
Lemma 2 If there is a unique value s∗ of s solving Q(s, Is) =λ, there is a unique equilibrium of the game in which every creditor rolls over the loan if and only if the signal exceeds the threshold s∗.
There are several ways to prove this threshold structure of the unique equilibrium, which is now familiar in the global games literature. The proof presented in appendix A follows Morris and Shin (1998). The crucial step of this (or any other) proof is to show that the decisions to roll over the loan are strategic complements, which, together with lemma 1, implies the result of lemma 2. Hence, lemma 1 and 2 together establish that there is a unique switching threshold s∗ and that the unique equilibrium strategy is Is∗(si). This in turn implies, together with equation (8), that there is a unique equilibrium thresholdθ∗(u)≡θcrit(s∗, u)for each stateu∈{u1, u2}such that the project succeeds if and only ifθ>θ∗(u).This proves proposition 1. Since the discrete threshold functionθ∗(u)takes only two different values, I will also use the notation θ∗(u1) =θ∗1 andθ∗(u2) =θ∗2. Note that while the threshold θ∗(u)depends on u,the strategy parameters∗ does not, foru is unknown at the time when the decision whether to roll over the loan is taken.
2.3 Equilibrium Thresholds
Given the unique equilibrium proÞle of strategies, we can solve the model for the equilibrium thresholds s∗, θ∗1 and θ∗2. First, Q(s∗, Is∗) = λ can be solved to obtain the equilibrium switching thresholds∗.Notice that in order to solve Q(s∗, Is∗) = λ for s∗, either the Þrst, second or third line of equation (10) has to be used, depending on the particular value of λ. In a second step, s∗ and equation (8) determine the two equilibrium thresholds θ∗1 andθ∗2,where one has to bear in mind that θ∗1 is simply u1 as soon ass∗ < u1−ε and that θ∗2 =u2+z ifs∗ > u2+z+ε. Summarizing the results, the model yields the following equilibrium thresholds
s∗ =
u2+λ2ε+z1−w −ε if 0<λ<λ wu1+ (1−w)u2+λ(2ε+z)−ε if λ≤λ ≤λ u1+ (λ+w−1)(2ε+z)w −ε if λ<λ<1
θ∗1 =
u1 if 0<λ<λ
2εu1+zu2+zw(u1−u2)
2ε+z +λz if λ≤λ ≤λ
u1+z³1−1−wλ´ if λ<λ<1
(11)
θ∗2 =
u2+ 1λz−w if 0<λ<λ u2+ zw(u2ε+z1−u2)+λz if λ≤λ ≤λ
u2+z if λ<λ<1
,
where the different ranges of λ are separated by λ= (1−w) [u1−u2]
2ε+z and λ= 1− w(u1−u2)
2ε+z . (12) It is important to stress that although the fundamentals uniquely de- termine whether the project succeeds, the outcome is still driven by the expectations of the agents. In many cases, creditors withdraw their loans only because they believe that others are going to do so, and since the other creditors share the same fear, a collapse of the Þrm can be the result of self- fulÞlling beliefs. If for example u1+ 2ε <θ <θ∗(u), which may well be the case if λ or z is large enough, the Þrm fails although everyone knows that this is not justiÞed by the fundamentals.11 Only for extreme values ofθ that lead to signals below s or above s for which agents have dominant actions, there is no need to care about other players’ beliefs. Therefore, the point I want to make is that the crucial role of the fundamental is not to exclude self-fulÞlling failures. Rather, given the assumed information structure, it uniquely determines the beliefs of the agents and thereby their behavior.
Another lesson of the preceding analysis is that the outcome of the model is not efficient, for if the creditors could coordinate their actions, they would agree to roll over their loans for more states of the economic fundamentals, thereby lowering the thresholds and the likelihood of a failure. In the Þrst best outcome, everyone would roll over her loan if and only if θ > u, which would require both coordination and the knowledge ofu. As a weaker conclu- sion, even ifuis unknown, there is no doubt that any fundamental threshold above u1 is inefficient.
11At the risk of raising confusion, one may even emphasize the subtle but essential role of higher order beliefs. A player may be reluctant to roll over even if she knows thatθ> u1 and that everyone knows that θ > u1, for she may not know that everyone knows that everyone knows thatθ> u1, which is in fact never common knowledge in this game. See Morris and Shin (1998), (2001) for more on higher order beliefs in global games.
In the remainder of this section, I discuss some properties of the thresholds θ∗1 and θ∗2, which in turn determine the likelihood of failure of the project.
AÞrst important result which follows easily from the thresholds in (11) and from u1 > u2 is that
θ∗1 >θ∗2, (13)
which simply means that the threshold above which the projects succeeds is higher in the worse case u =u1. Second, differentiating the thresholds with respect to the model parameters z andλ yields
∂θ∗(u)
∂z ≥0 and ∂θ∗(u)
∂λ ≥0. (14)
In economic terms, an increasing disruption z caused by early loan termi- nations raises ceteris paribus the critical threshold above which the Þrm survives. Also consistent with intuition, a growing λ, implying higher oppor- tunity costs of lending, increases the thresholds, since it makes the lenders more inclined to withdraw their funds before the project matures, which in term lowers the likelihood of success. The dispersion ε of the signal si, on the other hand, has no clear impact on the thresholds. More precisely,
∂θ∗1
∂ε ≥0 and ∂θ∗2
∂ε ≤0. (15)
Thus, an increasingε drives the two thresholds apart, if it has any inßuence at all. In order to analyze how u1 and u2 affect the thresholds, it seems useful to deÞne the numbers u andk such that u1 =u+k andu2 =u−k.
Differentiating the thresholds yields then
∂θ∗(u)
∂u = 1, (16)
whereas the sign of the derivative with respect to k is ambiguous. There is again an intuitive interpretation of the result in (16). Raising the average u of u1 and u2 corresponds to an adverse shift in the distribution of u, which is equivalent to a downward shift of the distribution of the Þrm speciÞc variable θ. Due to the assumed uniform distribution of θ, such a shift raises the thresholds by the same amount.
Finally, it is important to see that both thresholdsθ∗1andθ∗2 satisfy
∂θ∗(u)
∂w ≥0. (17)
Note that although θ∗1 =u1 or θ∗2 =u2+z for some values of λ,at least one of the two thresholds θ∗1 and θ∗2 is always strictly increasing in w. In other words, if the creditors attach a higher probability to the undesirable state u=u1, the project is more likely to fail.
With these results in mind, we are now ready to incorporate a second Þrm into the model and to explore the main implications of this extension.
3 The Model with 2 Firms
3.1 The Time Structure of the Model
In this part of the paper, I extend the model to include a secondÞrm. Thus, suppose there are two basically identical Þrms A and B both of which run a project Þnanced by two separate continua of creditors. Assume further that the state uis the same for bothÞrms, while there are two independent Þrm speciÞc states θA and θB. The latter can be interpreted either as two independent draws from one uniformly distributed fundamental variable or as the joint realization of two independent uniform random variables. The return of each Þrm is still given by equation (1), except that forÞrmA(Þrm B), θ is replaced byθA (θB).
In addition, I assume the following time structure, which is summarized in Þgure (2). In aÞrst stage, bothÞrms start their projects, and nature chooses the statesu,θAandθBaccording to the underlying probability distributions.
In a second stage, each creditor ofÞrmA receives a signal onθA as speciÞed in (3) and individually decides whether to withdraw her loan. Subsequently, the project ofÞrmAmatures and everyone, including the creditors ofÞrmB, observes whether Þrm A fails or not. However, neitheru nor θA is observed by anyone at this time. The agents may learn these realizations after the termination of projectBor even never, where the latter seems more plausible.
Having observed the outcome of A, the creditors of Þrm B receive a signal on θB and decide whether to withdraw their loans.12 Finally, the project of ÞrmB is completed.
Since the creditors ofAhave no additional information beyond the signal si when deciding on their loan, they behave like the creditors in the model
12They may also observe the signal earlier, e.g. together with the creditors ofÞrm A.
The crucial point is that the creditors ofÞrm B can wait longer until they have to decide whether they roll over their loans.
Figure 2: Timing of Events
with oneÞrm, and the equilibrium results of the previous sections go through unchanged. The creditors of the second Þrm, however, can observe the out- come of theÞrstÞrm. Does this observation provide useful information which affects the behavior of the creditors? Clearly, the answer is yes, as is shown in the next subsection.
3.2 Updating the Beliefs
To see what information can be gained from observing the outcome of A, considerÞrst what happens toÞrmA. Since theÞrstÞrm is equivalent to the singleÞrm in the model of section 2, except that the fundamentalθis replaced byθA,we know that there is a unique equilibrium in which the creditors ofA roll over their loans if and only if their signal exceeds some critical level s∗A, so that ÞrmA succeeds if and only if θA exceeds some threshold θ∗A(u). The thresholds of the signal and of θAare still given by the terms in (11), where w is the unconditional probability w=P(u= u1) =wa. In the following, I deÞne by θ∗A,1 andθ∗A,2 the two thresholds θ∗1 andθ∗2 computed withw=wa, which can be summarized in the discrete threshold function θ∗A(u) given by θ∗(u) with w = wa. Further, let FA denote the event where Þrm A fails andSA≡FA be the complementary event whereA survives. In equilibrium, ÞrmA fails whenever θA≤θ∗A(u). Thus, the formal deÞnitions of the events
”failure of A” and ”success of A” are:
FA = {{θA, u} |θA ≤θ∗A(u)} SA = {{θA, u} |θA >θ∗A(u)}.
Since θ is uniformly distributed over the unit interval, the probability that the realization of θ does not exceed a speciÞc threshold is equivalent to the
threshold itself, so it is straightforward to compute the following conditional probabilities:
P(FA|u=u1) =θ∗A,1 P(FA|u=u2) =θ∗A,2 P(SA|u=u1) = 1−θ∗A,1 P(SA|u=u2) = 1−θ∗A,2 .
(18)
In a next step, letwf ≡P(u=u1 |FA) denote the probability thatu is u1, conditional on the failure of A, while ws ≡ P(u = u1 | SA) shall denote the corresponding probability conditional on a success of A. It can then be shown that ws < wa < wf, which is an important result for what follows later. To prove this claim, notice that by Bayes rule, the probability of u being u1 conditional onFA can be computed as:
wf = P(FA|u=u1)P(u=u1)
P(FA|u=u1)P(u=u1) +P(FA |u=u2)P(u=u2).
Together with P(u = u1) = wa and P(u = u2) = 1−wa, equation (18) provides all probabilities that are needed to compute wf. Proceeding the same way, we can also compute the probability ws. The results in terms of the fundamental thresholds are:
wf = waθ∗A,1
waθ∗A,1+ (1−wa)θ∗A,2 (19) ws = wa(1−θ∗A,1)
1−waθ∗A,1−(1−wa)θ∗A,2. (20) Finally, notice from inequality (13) in section 2.3 that θ∗A,1 > θ∗A,2, which implies
θ∗A,1 > waθ∗A,1+ (1−wa)θ∗A,2 and, after multiplying both sides by waθ∗A,1+(1wa−wa)θ∗A,2,
wf = waθ∗A,1
waθ∗A,1 + (1−wa)θ∗A,2 > wa. (21) A symmetric argument establishes that
ws = wa(1−θ∗A,1)
1−waθ∗A,1−(1−wa)θ∗A,2 < wa. (22)
Together, (21) and (22) prove that ws < wa < wf as claimed. Hence, it is rational (and quite intuitive) that the creditors of ÞrmB consider the worse state u=u1 to be more likely when they observe that Afails. Though such failure can be due to a low θA and not due to a high realization of u, it provides additional information (i.e. bad news) on the true state of u. On the other hand, conditional on a success of project A, the event u = u1 is less likely than unconditionally.
3.3 Failure Probabilities and Contagion
What is now going to happen to the second Þrm? From the discussion in the previous section, we know that the only difference between the creditors of the two Þrms lies in their different assessment of the state u. The outcome of the Þrst project serves as a signal which leads to an updated information on the probability distribution of u. Since this signal is commonly observed, all creditors of B share the same updated beliefs over u, and the only con- sequence for their decision making is that the probability w is either ws or wf. Therefore, the unique equilibrium of the game among the creditors of B is still obtained as in section 2.2, with the important difference that the equilibrium depends on the outcome of Þrm A. If the Þrst Þrm fails, the two fundamental thresholds of θB can be calculated as in (11) with w=wf, whereas if A succeeds, the thresholds ofθB are computed withw=ws.
In the following, letθfB,j (respectivelyθsB,j) denote the thresholds ofθB if A fails (respectively succeeds), wherej ∈{1,2}captures the state ofu. The thresholds ofθB can also be written as a function θ∗B(u,θA),whereθ∗B(u,θA) is θ∗(u) computed with w = wf if θA ≤ θ∗A(u) and θ∗B(u,θA) = θ∗(u) with w = ws if θA > θ∗A(u). Further, deÞne by FB and SB ≡ FB the events where ÞrmB fails respectively survives. Failure of B occurs whenever θB ≤ θ∗B(u,θA), i.e. if A fails and θB ≤ θfB,j or if A succeeds and θB ≤ θsB,j for j ∈ {1,2}.Formally, the two mutually exclusive and exhaustive setsFB and SBare deÞned as:
FB = {{θA,θB, u} |θB≤θ∗B(u,θA)} SB = {{θA,θB, u} |θB>θ∗B(u,θA)}
The important point to see is that the threshold of θB depends not only on u, but also on θA, althoughθA is independent of any economic fundamental which affects Þrm B.
Since θB is uniformly distributed over the unit interval, the thresholds of θB represent at the same time conditional failure probabilities, which are summarized in the following equation:13
P (FB |(u=u1)∩FA) =θfB,1 P (FB |(u=u1)∩SA) =θsB,1 P (FB |(u=u2)∩FA) =θfB,2 P (FB |(u=u2)∩SA) =θsB,2.
(23)
Furthermore, the failure probability of B conditional on failure of A can be written as
P(FB |FA) = P (FB |(u=u1)∩FA)P (u=u1 |FA) +P (FB |(u =u2)∩FA)P(u=u2 |FA),
which, after plugging in the results of (19) and (23), yields the probability P(FB |FA) = waθ∗A,1θfB,1+ (1−wa)θ∗A,2θfB,2
waθ∗A,1+ (1−wa)θ∗A,2 . (24) Accordingly, if the creditors of the second Þrm observe that A is successful, ÞrmB fails with probability
P(FB |SA) = wa(1−θ∗A,1)θsB,1+ (1−wa)(1−θ∗A,2)θsB,2
1−waθ∗A,1−(1−wa)θ∗A,2 . (25) The probabilities P(SB | FA) and P(SB | SA) of a success of Þrm B, con- ditional on either the failure or survival of A, are calculated analogously, or simply as 1−P(FB |FA) and1−P(FB |SA).
Finally, we can also calculate theunconditional failure probability of each Þrm. First, note that the likelihood of failure of A can be written as
P(FA) =P(FA |u=u1)P(u=u1) +P(FA|u=u2)P(u=u2)
Therefore, plugging inP(u=u1) =wa,P(u=u2) = 1−waand the relevant probabilities from (18), ÞrmA fails with probability
P(FA) =waθ∗A,1+ (1−wa)θ∗A,2 (26)
13Naturally, the corresponding survival probabilities ofÞrmB are easily found by sub- stracting the relevant threshold from one.
and survives with probability P(SA) = 1−P(FA).The unconditional prob- ability of failure of the second Þrm can then be computed as
P(FB) =P(FB |FA)P(FA) +P(FB |SA)P(SA).
which, using equations (26), (24) and (25), results in
P(FB) = wahθ∗A,1θfB,1+ (1−θ∗A,1)θsB,1i (27) +(1−wa)hθ∗A,2θfB,2+ (1−θ∗A,2)θsB,2i.
The unconditional probability that B survives, of course, is then given by P(SB) = 1−P(FB).
To complete the discussion of this section, it is worthwhile having a closer look at the thresholds of ÞrmB conditional on the outcome of theÞrstÞrm.
From the crucial result (17) in section 2.3, we know that the threshold θ∗(u) is increasing in w for at least one state of u. Together with wf > ws, this implies that
θsB,1 ≤θfB,1 (28)
and
θsB,2 ≤θfB,2, (29)
where at least one of the inequalities (28) and (29) holds strictly. This in turn proves that the failure of Aaffects the behavior of the creditors ofB in a way which increases the thresholds and thereby the likelihood of failure of B. By the same mechanism, however, a success of A has a positive impact on B. The following proposition summarizes these important Þndings.
Proposition 2 (Contagion) If the creditors of Þrm B observe the outcome of ÞrmA before deciding on their loan, there exists (for at least one state u) a range of θB for which B fails if A fails and succeeds if A survives. Hence, a failure of A increases the likelihood of failure of Þrm B, whereas a success of Þrm A makes a failure of Þrm B less likely.
Proposition 2 is a crucial result, establishing that there is a contagious link between the twoÞrms such that the probability thatB fails is higher ifA fails than conditional on a success ofA. While theÞrst part of the proposition follows directly from the fact that at least one of the inequalities (28) and (29) holds strictly, the second statement which concerns the likelihood of failure may require some additional explanations, for which I refer to the next subsection.
3.4 Contagion and Correlation
According to proposition 2, the failure of A increases the probability that B fails. However, this is a potentially misleading statement which deserves closer attention. Of course, because of their joint dependence on the common stateu, the outcomes of the twoÞrms are not uncorrelated even without con- tagion, but the point I want to make is that contagion leads to an additional increase of this ”natural” correlation. To see this, let me sketch the following (and hopefully not confusing) reasoning. DeÞne by Pe(FB | FA) the proba- bility that B fails conditional on the failure of A, but under the assumption that there is no contagion. The latter assumption implies that the thresh- olds of B are left unchanged and correspond to those of A. Accordingly, let Pe(FB |SA) be the likelihood thatB fails provided thatA succeeds and that there is no contagion. In other words, we might think of the probabilities from the perspective of an outsider who can observe the outcome of A, while the creditors of B cannot. Given these deÞnitions, I show in appendix B.1 that the following chain of inequalities holds:
P(FB |FA)>Pe(FB |FA)>Pe(FB |SA)> P(FB |SA). (30) Thus, compared to the unconditional likelihood of failure, the conditional probability that B fails is higher if A fails than if A succeeds even without contagion. But if there is contagion, inequality (30) shows that the con- ditional failure probability is further increased if A fails and lowered if A succeeds. The difference between P(FB | FA) and Pe(FB | FA) or between Pe(FB | SA) and P(FB | SA) captures the effect on the failure probability which stems from the change in the behavior of investors.
The Þrst part of the correlation which is caused by common (macroeco- nomic) fundamentals should not be mixed up with contagion. Taking the banking industry as an example, it may happen that several banks crash simultaneously because they jointly face extremely bad economic conditions, which need not mean that they have contagiously affected each other. In my view, a reasonable understanding of Þnancial contagion can only include forces which originate in Þnancial markets (e.g. changes in the behavior of investors) and which remain effective even if one controls for any common third inßuencing factors. In the present model, contagion implies that even conditional on the state u, the likelihood that B fails is higher (lower) if A fails (succeeds). In this sense, it is correct to conclude that by altering the behavior of creditors, a collapse of A cancause the failure of B.
3.5 The Model with N Þrms
So far, there are only two Þrms in the model. Before turning to a rigorous discussion of the consequences of contagion in section 4, I would like to men- tion that in principle, the model is easily generalized to includeN ≥2Þrms.
In a Þrst step, we can introduce a third Þrm C which is equivalent to A or B, except that its investors can observe the outcome of bothAandB before having to decide on their loans. Proceeding analogously, we can include ar- bitrarily manyÞrms. The imposed time structure implies that every project has a contagious impact on all projects which follow later in time but on none which matures earlier. This framework raises the possibility that the failure of one or a fewÞrms can trigger a chain of failures and may even ruin all otherÞrms, which is however an extremely unlikely event. Moreover, this is again only one side of contagion, because a single successful Þrm might as well rescue the other Þrms. Since the basic mechanisms and insights of the model with twoÞrms remain unchanged, whereas the Bayesian updating process becomes already involved with three Þrms, a thorough discussion of this generalization is beyond the scope of this paper.
4 Contagion Revisited
4.1 Positive versus Negative Contagion
A major result of the model with two (or more) Þrms is that there exists Þnancial contagion, such that the failure of Þrm A can trigger the collapse of ÞrmB. Thus, the outcome of project B depends on θA even though the Þrm speciÞc fundamental variables θA and θB are independent. There is also no direct connection in the sense that B is a creditor, a supplier or a customer ofA. The contagious link between the twoÞrms stems from a purely informational mechanism, which arises because the creditors of Þrm B can observe whether A survives or not. Based on this signal, they (reasonably) adjust their beliefs on the state of the fundamental u, which in turn affects the thresholds of B in such a way that projectB is more likely to share the same fate as projectA. In this section, I discuss the basic mechanism and the consequences of contagion in some more detail. The following discussion is meant to provide a better understanding of contagion and on the conclusions which can be drawn, since these may prove less obvious than suggested in the existing literature.
Given the result in proposition 2, one might take the difference between the thresholds θfB,j andθsB.j or the probabilities P(FB |FA) andP(FB |SA) as a measure to quantify the negative impact of contagion and jump to the conclusion that this would be the gain from eliminating contagion. However, θsB,j and P(FB | SA) are not necessarily the appropriate benchmarks with which θfB,j and P(FB |FA) should be compared when discussing the impact of contagion. Rather, the outcome of the model should be compared to a scenario in which there is no Þnancial contagion. One approach to get rid of the contagious link is to suppose that the creditors of B cannot observe the outcome of project A before they have to take action. Equivalently, we could assume that two different states ua and ub are relevant in the return function (1) of Þrm A and B, where ua and ub are independent. A third benchmark scenario is simply to assume that ÞrmB is the only presentÞrm.
Each of these three scenarios provides a reasonable benchmark case in which the relevant thresholds of ÞrmB are the same as those of A.
Taking this hypothetical scenario of an ”autark”Þrm as a starting point, we can apply the following deÞnition of contagion: Contagion occurs when- ever the outcome of projectBdeviates from the outcome which would prevail in the benchmark case. If the second Þrm fails only because of the adverse signal sent out by the collapse of A, the owner and the lenders ofÞrmB are apparently worse off because ofA. Therefore, I deÞne this event as ”negative contagion”. On the other hand, there is also an event where Þrm B is suc- cessful but would not be so in the benchmark case. At the risk of depriving the literature on contagion of its only agreement, I deÞne this event as ”- positive contagion”. Since contagion is typically perceived as an undesirable phenomenon, these deÞnitions may prove to be controversial. Those readers who insist on contagion being a negative event per deÞnition may regard the term ”negative contagion” as a pleonasm, while others mayÞnd the notion of
”positive contagion” hard to accept. Of course, this is a matter of deÞnition, but it cannot be overemphasized that whatever name we choose for the two phenomena, they occur by the same mechanism.
This does by no means contradict the traditional argument of the in- formational channel that the failure of one Þrm reveals adverse information on other Þrms. To be sure, there is nothing wrong with this explanation.
However, if we accept this argument, then the success of the very same Þrm must also reveal some information, which must lead to an adjustment of the players’ priors in the opposite direction. To see this, consider the following
